the ball pivoting algorithm

The Ball Pivoting Algorithm

Ioannis Stamos

Points vs. Meshes

Point-based vs. Mesh-based representation.+ Fast rendering for complicated point-sets.+ No loss of information (all points used).+ No need for modeling of every tiny scene detail.- Rendering quality degrades when viewer zooms-in.- Computation waste in mainly planar areas.- Solid modeling operations not straightforward- Not supported in commercial modeling/rendering software

(Maya, etc.)

Meshing Algorithms

• Input: Range Images– Range Image = Set of 3D points

• Output: Single mesh = topologically correct set of triangles (not triangle soup)

Range Sensor Imaging Characteristics

Range Image:rectangular sampling

Image has structure surface elements occlusion elements

From Points to Mesh [Single Image]

Sensing Direction

Discard long edges ?Threshold distance ?

Occlusion

3D-modeling pipeline

3D-to-3D Registration

Meshing

Remove redundancy

Aligned meshes

Single Mesh

Range ImageAcquisition

Methodology• Each range image is preprocessed

3D points P => 3D points with associated normals: (P, N)

• Bounding box enclosing all images computed

• Ball Pivoting Algorithm:– Input: 3D points and normals from all range images– Output: Triangular mesh

N

P

•Patch fit in K by K neighborhood around point P•Eigenvector corresponding to smallest eigenvalue: normal•Magnitude of smallest eigenvalue: confidence in normal•Normal computed for every 3D point•Structure of range images used for K by K neighbourhood-1

Preprocessing Range Images

K by K neighbourhood around point P

Neighborhood used for normal computation

Accurately computednormals in locally

planar regions

Neighborhood used for normal computation

Two possibledirections

[towards outside]

Neighborhood used for normal computation

Two possibledirections

[towards inside]

Neighborhood used for normal computation

Consistent normaldirection

for all points

Neighborhood used for normal computation

Consistent normaldirection

for all points

Choose outsideorientation

Neighborhood used for normal computation

Inaccurately computednormals

close to boundaries

Ball Pivoting Algorithm [Input]

• Points/normals of all range images in data structure• Connectivity of points in range images not needed anymore• Radius ρ selected by the user.

Voxel-based data structure

x-axis

y-axis

ρ

ρ

Fast search for points within

circle of radiusρ

Voxel-based data structure

x-axis

y-axis

ρ

ρ

Fast search for points within

circle of radiusρ

Need to search only inadjacent voxels

[9 in this picture –27 in 3D space]

Voxel-based data structure

x-axis

y-axis

ρ

ρ

Fast search for points within

circle of radiusρ

Need to search only inadjacent voxels

[9 in this picture –27 in 3D space]

Basic OperationsF. Bernardini, J. Mittleman, H. Rushmeier, C. Silva, G. Taubin.

The ball-pivoting algorithm for surface reconstruction.

IEEE Trans. on Vis. and Comp. Graph. 5 (4), pages 349-359, October-December 1999.

A sequence of ball-pivoting operations. From left to right: A seed triangle is found; pivoting around an edge of the current front adds a new

triangles to the mesh; after a number of pivoting operations, the active front closes on itself; a final ball-pivoting completes the mesh.

Closely related to alpha-shapes, Edelsbrunner 94

Seed triangles

A number of seed triangles with their associated spheres shown

Pivoting in 2D

(a) Circle of radius ρ pivots from point to point, connecting them with edges.

(b) When sampling density is low, some of the edges will not be created, leaving holes.

(c) When the curvature of the manifold is larger than 1/ρ, some of the points will not

be reached by the pivoting ball, and features will be missed.

The algorithm [Edge representation]

• Edge (si, sj)– Opposite point so, center of empty ball c– Edge: “Active”, “Boundary”, or “Frozen”

sisj

so

c

Pivoting example

Active edge

Point on front

Initial seed triangle:Empty ball of radius ρ passes through the three points

Pivoting example

Active edge

Point on frontBall pivoting around active edge

Pivoting example

Active edge

Point on frontBall pivoting around active edge

Pivoting example

Active edge

Point on frontBall pivoting around active edge

Pivoting example

Active edge

Point on frontBall pivoting around active edge

Pivoting example

Active edge

Point on frontInternal point

Ball pivoting around active edge

Pivoting example

Boundary edge

Point on frontInternal point

Ball pivoting around active edgeNo pivot found

Active edge

Pivoting example

Point on frontInternal point

Active edge

Ball pivoting around active edge

Boundary edge

Pivoting example

Point on frontInternal point

Active edge

Boundary edge

Ball pivoting around active edgeNo pivot found

Pivoting example

Point on frontInternal point

Active edge

Boundary edge

Ball pivoting around active edge

Pivoting example

Point on frontInternal point

Active edge

Boundary edge

Ball pivoting around active edge

Frozen edge

Points in frozen region

Pivoting example

Point on frontInternal point

Active edge

Boundary edge

Ball pivoting around active edge

Frozen edge

Points in frozen region

Pivoting example

Point on frontInternal point

Active edge

Boundary edge

Ball pivoting around active edge

Frozen edge

Points in frozen region

Pivoting example

Point on frontInternal point

Active edge

Boundary edge

Ball pivoting around active edge

Frozen edge

Points in frozen region

Algorithm Snapshot

• 13 points (P1, …)• 4 voxels (V0, …)• 11 edges (E1, …)• Two loops: L1, L2

Noisy data

(a) Points “below” surface level are not touched by the pivoting ball and remain isolated (and are discarded by the algorithm).

(b) Due to missing data, the ball pivots around an edge until it touches a sample that belongs to a different part of the surface. By checking that triangle and data point normals are consistently oriented, we avoid generating a triangle in this case.

(c) Noisy samples form two layers, distant enough to allow the ball to “walk” on both layers. A spurious small component is created.

Ball Pivoting Algorithm

Overlapping scans

3D mesh

Ball Pivoting Algorithm

3D mesh detail 3D mesh detail

Out of core implementation• Process data in slices that fit in

memory• No limit in size of input• BPA’s active front provides

natural implementation• Of major importance for large

scale scenes

Out of core implementation

Bounding box of scene

Out of core implementation

Bounding box of sceneFirst slice in memory

Mesh forfirst slice

Frozen region

Out of core implementation

Second slice in memory

Mesh forsecond slice

Frozen region

Out of core implementation

Third slice in memory, etc.

Mesh forthird slice

Frozen region

Out of core results

Part of the Great Hall mesh. The different colors correspond to meshes of different slicesproduced by the out-of-core implementation.

Results

• Grand Central Station– Leica 4500 (phase-based)

• 27 input scans ~ 40 million points

• Split into 30 slices

• Preprocessing ~ 40 minutes

• Split operation ~ 37 minutes

• BPA out-of-core ~ 22 hours

[ Intel Xeon Processor at 2GHz, 2Gb of RAM]

• Output mesh ~ 7.95 million triangles / 4.19 million vertices

• Ball radius ρ = 0.2 meters ( ~ 7.87 inches)

Results

• Grand Central Station– Cyrax 2500 (time of flight)

• 45 input scans ~ 45 million points• Split into 50 slices• Normal computation ~ 38 mins• Split operation ~ 65 mins• BPA out-of-core ~ 18 hours[ Intel Xeon Processor at 2GHz, 2Gb of RAM]• Output mesh ~7.74 million triangles / 3.98 million vertices• Ball radius ρ = 0.1 meters ( ~ 3.93 inches)

Results

• Great Hall– Cyrax 2500 (time of flight)

• 27 input scans ~ 27 million points• Split into 30 slices• Normal computation ~ 20 mins• Split operation ~ 27 mins• BPA out-of-core ~ 3 hours[ Intel Xeon Processor at 2GHz, 2Gb of RAM]• Output mesh ~ 20.40 million triangles / 11 million vertices• Ball radius ρ = 0.03 meters ( ~ 1.18 inches)

Out of core result [Great Hall]

Out of core result [Grand Central]

Out of core result [Grand Central]

Out of core result [Great Hall]

Out of core result [Great Hall]

Out of core result [Great Hall]

Out of core result [Great Hall]

Out of core result [Grand Central]

Out of core result [Grand Central]

Out of core result [Grand Central]

Advantages

• Efficient and conceptually natural algorithm• Provides topologically correct mesh• Faithfully follows data• Models scenes of any geometric type• Models scenes of any size (out of core)• Larger ball : cruder / smaller model • Smaller ball: sharper / larger model [captures detail]

Disadvantages

• Ball radius ρ : does not adapt to local point density – Can use balls of increased radii [not a good solution]

• Sensitive to noisy normals– Smoothing of normals may help as pre-processing step

• Faithfully follows data– May need to smooth mesh as post-processing step

• Holes are generated– Small holes due to noisy normals or variable point density– Large holes in areas containing no data– A hole filling algorithm is essential [future work]